Explore a groundbreaking approach to the Breuil-Mezard Conjecture in this hour-long lecture by Tony Feng at the Hausdorff Center for Mathematics. Delve into the concept of 'Breuil-Mezard cycles' in the moduli space of mod p Galois representations of Qp and their role in governing congruences between mod p automorphic forms on reductive groups. Examine the connection to the weight part of Serre's Conjectures for GL₂. Discover a novel, purely local and group-theoretic method developed in collaboration with Bao Le Hung, which employs geometric representation theory and microlocal analysis instead of traditional patching techniques. Investigate the intriguing links between the Breuil-Mezard Conjecture and advanced concepts in quantum groups and homological mirror symmetry, offering a fresh perspective on this fundamental problem in number theory and representation theory.